Almost minimizers for a sublinear system with free boundary

TL;DR

This paper investigates the regularity of vector-valued almost minimizers for a sublinear energy functional with free boundary, using epiperimetric inequalities to analyze the structure and smoothness of solutions and free boundary points.

Contribution

It introduces a novel approach employing epiperimetric inequalities to establish regularity results for almost minimizers in a sublinear free boundary problem.

Findings

Proved regularity of almost minimizers with Hölder continuous coefficients.

Characterized the set of regular free boundary points.

Established smoothness properties of the free boundary.

Abstract

We study vector-valued almost minimizers of the energy functional $$\int_D\left(|\nabla\mathbf{u}|^2+\frac2{1+q}\left(\lambda_+(x)|\mathbf{u}^+|^{q+1}+\lambda_-(x)|\mathbf{u}^-|^{q+1}\right)\right)dx,\quad0<q<1.$$ For H\"older continuous coefficients $\lambda_\pm(x)>0$, we take the epiperimetric inequality approach and prove the regularity for both almost minimizers and the set of "regular" free boundary points.